I've never seen Peaky Blinders, but with this question and a previous question concerning a depiction of a black voter in the same series, I feel really tempted to see it because this is really paying close attention to historical details.
First, a little context. Peaky Blinders takes place in Birmingham, one of the many ports and cities in Great Britain to be closely connected to the transatlantic economy, the triangular trade and colonialism throughout the 18th, 19th and 20th century. This made Birmingham a natural destination for men and women of African descent who throughout these centuries arrived as sailors, students, workers, slaves, or simply migrants looking for a life in the "mother country". By the time of the mid-1920s, there were domiciled black British families who had lived in Great Britain for generations. There were also many men of African descent from all corners of the British Empire who had arrived during the First World War as a consequence of the war. It is against this background that we need to contextualize the lone black British Colonel in S04E05 of Peaky Blinders.
According to the King's Regulation, the rules are very clear: No men of non-European ancestry can be commissioned into the British Army. The 1907 Manual of Military Law does allow British men of non-European ancestry to enter the British Army (and many of them did!), but the King's Regulation were far more explicit. Does this mean that there were no black British officers during the First World War and into the 1920s? The answer there is also simple: No. Rules (and regulations) are meant to be broken and in the case of the British Army in the First World War, they certainly were.
The most famous case is Second Lieutenant Walter Tull (1888-1918) and for many decades, he became the first and only black British officer during the First World War in popular knowledge. But recently, groundbreaking research by historian Ray Costello has unearthed more officers of African descent in the British Army during the time period in question.
Before I go any further, let's talk about sources. A big obstacle for any researcher dealing with minorities in Europe (unlike the United States) is that the topic of race is not commonly mentioned on the rosters, draft cards or other military documents that attests to an individual's service. Many, if not all, domiciled Black British soldiers did not have any names that would make them stick out of a crowd when you're going through lists looking for them. You could sometimes be in luck if you manage to find a medical note commenting on the soldier's race, usually in a negative form, but this isn't a frequent occurrence. Many researchers today investigating black British soldiers in the First and Second World War have to base their research on matters like local and family history, contemporary and post-war reports of black British servicemen in newspapers and other popular media as well as the occasional mentions in official government and military documentation. This particular area remains very understudied and it's only recently that historians have begun writing on the topic of black British soldiers in the First World War. This also makes it very difficult right now to make a proper judgement on just how many black British officers there were during the First World War and the 1920s and just how common this occurrence would have been. What we do know is that it would be realistic for a Black British officer to find himself in Birmingham in the 1920s.
As Ray Costello writes, the "interpretation of a recruiting officer or an interviewing board could determine a man's acceptance into the ranks, or whether he was offered a commission." This meant that young black men could be turned away from joining the British Army on the account of their race, some medical examiners judging them as being "disabled" due to their race, but it also meant that against all odds, young black men could become commissioned into the British Army by a commissioning officer turning a blind eye for example. Other factors, such as high social status or being a light-skinned individual made it easier to get a commission. Black British officers like Second Lieutenant George Edward Kingsley Bemand (1892-1916), Second Lieutenant John Albert Gordon Smyth (1886-1918), Second Lieutenant Reginald Emanuel Collins (1894-?), Captain (later Brigadier General) Horace Somerville Sewell (1881-1953), and Lieutenant David Louis Clemetson (1893-1918) are some of the black British officers that we are aware of at the present time. Unfortunately, as you can see, many of them lost their lives in the Great War but others, like Horace Somerville Sewell, survived and made a career out of the British Army. Horace Somerville Sewell is particularly interesting for serving throughout the entire war (1914-1918) and being awarded the DSO with bar, the French Légion d'honneur and a CMG.
This list of black British officers is a work in progress. Researchers are constantly finding new names and new lives to add to this list, and I hope that there are young and eager researchers out there who are interested in the experiences of minorities during the First World War who might put us all to shame with their amazing research. More research needs to be done about black British participation in the British Army during the interwar years and would make a great topic for a potential dissertation. With that said, the King's regulations wouldn't change until 1940 and a new war. We have the British civil rights organization League of Coloured Peoples to thank for putting that change into motion.
Source:
Black Tommies: British Soldiers of African Descent in the First World War by Ray Costello (University of Liverpool Press, 2016).